Theorem nelsn

Description: If a class is not equal to the class in a singleton, then it is not in the singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020) (Proof shortened by BJ, 4-May-2021)

	Ref	Expression
Assertion	nelsn	$$⊢A\ne B\to \neg A\in \left\{B\right\}$$

Proof

Step	Hyp	Ref	Expression
1		elsni	$$⊢A\in \left\{B\right\}\to A=B$$
2	1	necon3ai	$$⊢A\ne B\to \neg A\in \left\{B\right\}$$